Microstrip Antenna – Cavity Model

The following is an alternative modelling technique for the microstrip antenna, which is also somewhat similar to the analysis of acoustic cavities. Like all cavities, boundary conditions are important. For the microstrip antenna, this is used to calculated radiated fields of the antenna.

Two boundary conditions will be imposed: PEC and PMC. For the PEC the orthogonal component of the E field is zero and the transverse magnetic component is zero. For the PMC, the opposite is true.

cavity

This supports the TM (transverse magnetic) mode of propagation, which means the magnetic field is orthogonal to the propagation direction. In order to use this model, a time independent wave equation (Helmholtz equation) must be solved.

helmholtz

The solution to any wave equation will have wavelike properties, which means it will be sinusoidal. The solution looks like:

1234

Integer multiples of π  solve the boundary conditions because the vector potential must be maximum at the boundaries of x, y and z. These cannot simultaneously be zero. The resonant frequency can be solved as shown:

res

The units work out, as the square root of the product of the permeability and permittivity in the denominator correspond to the velocity of propagation (m/s), the units of the 2π term are radians and the rest of the expression is the magnitude of the k vector or wave number (rad/m). This corresponds to units of inverse seconds or Hz. Different modes can be solved by plugging in various integers and solving for the frequency in Hz. The lowest resonant mode is found to be f_010 which is intuitively true because the longest dimension is L (which is in the denominator). The f_000 mode cannot exist because that would yield a trivial solution of 0 Hz frequency. The field components for the dominant (lowest frequency) mode are given.

1x

 

 

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